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manuscripta mathematica

, Volume 94, Issue 1, pp 169–186 | Cite as

Ons-sets and mutual absolute continuity of measures on homogeneous spaces

  • Tord Sjödin
Article

Abstract

We extend a recent result of A. Jonsson about mutual absolute continuity of twoD s -measures on ans-setFR n to the homogeneous spaces (X, d, μ) of Coifman, Weiss. Here we define Hausdorff measure, Hausdorff dimension,D s -set andd-set relative to the measureμ. Our main result holds for so called (s, d)-sets,ds, and is stronger than Jonssons result even inR n . As applications we interpret this Hausdorff dimension as a relative dimension for very regular sets and show that it in general depends strongly onμ. For this purpose we construct a strictly increasing functionf :RR, whose measure is doubling and concentrated on a set of arbitrary small Hausdorff dimension. The extension off to a quasiconformal map of the half plane onto itself sharpens a classical example of Ahlfors-Beurling.

Keywords

Homogeneous Space Hausdorff Dimension Besov Space Quasiconformal Mapping Hausdorff Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Tord Sjödin
    • 1
  1. 1.Department of MathematicsUniversity of UmeåUmeåSweden

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